Embed to rigorously and accurately homogenise quasi-periodic multi-scale heterogeneous PDEs, with computer algebra

Abstract

For microscale heterogeneous PDEs, this article further develops novel theory and methodology for their macroscale mathematical/asymptotic homogenization. This article specifically encompasses the case of quasi-periodic heterogeneity with finite scale separation: no scale separation limit is required. Dynamical systems theory frames the homogenization as a slow manifold of the ensemble of all phase-shifts of the heterogeneity. Depending upon any perceived scale separation within the quasi-periodic heterogeneity, the homogenization may be done in either one step, or two sequential steps: the results are equivalent. The theory not only assures us of the existence and emergence of the homogenization, it also provides a practical systematic method to construct the homogenization to any specified order. For a class of heterogeneities, we show that the macroscale homogenization is potentially valid down to lengths which are just twice that of the microscale heterogeneity! This methodology provides a new rigorous and flexible approach to homogenization that potentially also provides correct initial and boundary conditions, treatment of forcing and control, and analysis of uncertainty.